Anderson Acceleration Without Restart: A Novel Method with $n$-Step Super Quadratic Convergence Rate

TL;DR

A novel Anderson's acceleration method to solve nonlinear equations, which does not require a restart strategy to achieve numerical stability and has an $n$-step super quadratic convergence rate, where $n$ is the dimension of the objective problem.

Abstract

In this paper, we propose a novel Anderson's acceleration method to solve nonlinear equations, which does \emph{not} require a restart strategy to achieve numerical stability. We propose the greedy and random versions of our algorithm. Specifically, the greedy version selects the direction to maximize a certain measure of progress for approximating the current Jacobian matrix. In contrast, the random version chooses the random Gaussian vector as the direction to update the approximate Jacobian. Furthermore, our algorithm, including both greedy and random versions, has an $n$-step super quadratic convergence rate, where $n$ is the dimension of the objective problem. For example, the explicit convergence rate of the random version can be presented as $ \norm{\vx_{k+n+1} - \vx_*} / \norm{\vx_k- \vx_*}^2 = \cO\left(\left(1-\frac{1}{n}\right)^{kn}\right)$ for any $k\geq 0$ where $\vx_*$ is the optimum of the objective problem. This kind of convergence rate is new to Anderson's acceleration and quasi-Newton methods. The experiments also validate the fast convergence rate of our algorithm.

Figures (1)

• Figure 1: Numerical experiments.

Theorems & Definitions (33)

• Definition 2.1: $n$-step super quadratic convergence
• Definition 2.4
• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 4.1